Hi, learning about homemorphic functions at the moment and am failry confused,
If i have a homeomorphic function h: A - B between two topological spaces A and B,
Is the pre image of the empty set in B the empty set in A ?
Look closely at the definition of pre-image: If $\displaystyle f:A\to B$ and $\displaystyle V\subseteq B,$ the pre-image of $\displaystyle V$ under $\displaystyle f$ is $\displaystyle f^{-1}(V)=\{x\in A:f(x)\in V\}.$ In other words, $\displaystyle x\in f^{-1}(V) \Leftrightarrow f(x)\in V$ for all $\displaystyle x\in A.$ Now try assuming $\displaystyle f^{-1}(\O)\ne\O$ and see what happens.